Question: Find $x.$

[asy]
unitsize(0.7 cm);

pair A, B, C, D, O;

O = (0,0);
A = 4*dir(160);
B = 5*dir(160 + 180);
C = 8*dir(20);
D = 4*dir(20 + 180);

draw(A--B);
draw(C--D);
draw(A--C);
draw(B--D);

label("$4$", (A + O)/2, SW);
label("$10$", (C + O)/2, SE);
label("$4$", (D + O)/2, NW);
label("$5$", (B + O)/2, NE);
label("$8$", (B + D)/2, S);
label("$x$", (A + C)/2, N);
label("$A$", A, W);
label("$B$", B, E);
label("$C$", C, E);
label("$D$", D, W);
label("$O$", O, N);
[/asy]
Answer: Let $\theta = \angle AOC = \angle BOD.$  Then by the Law of Cosines on triangle $BOD,$
\[\cos \theta = \frac{4^2 + 5^2 - 8^2}{2 \cdot 4 \cdot 5} = -\frac{23}{40}.\]Then by the Law of Cosines on triangle $AOC,$
\begin{align*}
x^2 &= 4^2 + 10^2 - 2 \cdot 4 \cdot 10 \cos \theta \\
&= 4^2 + 10^2 - 2 \cdot 4 \cdot 10 \cdot \left( -\frac{23}{40} \right) \\
&= 162,
\end{align*}so $x = \sqrt{162} = \boxed{9 \sqrt{2}}.$